CHAPTER
Nonparametric Tests
11
11.1 THE SIGN TEST
11.1 Try It Yourself Solutions
a
b. 0.025
α
=
c. n = 24
d. The critical value is 6.
a
b. 0.10
α
=
c. n = 92
d. The critical value is 01.645.z=−
e. x = 41
z = ( 0.5) 0.5( ) (41 0.5) 0.5(92) 4.5 0.94
4.80
92
22
xn
n
+= +− −
=≈≈−
:
a
b. 0.05
α
=
c. n = 11
d. The critical value is 2.
454 CHAPTER 11 │ NONPARAMETRIC TESTS
11.1 EXERCISE SOLUTIONS
1. A nonparametric test is a hypothesis test that does not require any specific conditions concerning
the shapes of populations or the values of any population parameters.
A nonparametric test is usually easier to perform than its corresponding parametric test, but the
nonparametric test is usually less efficient.
2. Median
3. When n is less than or equal to 25, the test statistic is equal to x (the smaller number of + or −
4. Answers will vary. Sample answer: It is called the sign test because each value in a sample is
5. Identify the claim and state 0
H and a
H. Identify the level of significance and sample size. Find
the critical value using Table 8 (if 25n≤) or Table 4 (n > 25). Calculate the test statistic. Make a
decision and interpret in the context of the problem.
6. A sample must be randomly selected from each population and the samples must be dependent.
7. (a) The claim is “the median amount of new credit card charges for the previous month was more
than $300.”
8. (a) The claim is “the median daily high temperature for the month of July in Pittsburgh is 83°
Fahrenheit.”
0:H median = 83 (claim); :
a
H median ≠ 83
CHAPTER 11 │ NONPARAMETRIC TESTS 455
9. (a) The claim is “the median sales price of new privately owned one-family homes sold in the
past year is $198,000 or less.”
0:H median ≤ $198,000; :
a
H median > $198,000 (claim)
10. (a) The claim is “the median daily high temperature for the month of January in San Diego is 66°
Fahrenheit.”
0:H median = 66 (claim); :
a
H median ≠ 66
(b) The critical value is 2.
(c) x = 0
(d) Reject
0.H
(e) There is enough evidence at the 1% level of significance to reject the Meteorologist’s claim
that the median daily high temperature for the month of January in San Diego is 66°
Fahrenheit.
12. (a) The claim is “the median amount of financial debt for families holding such debt is less than
$65,000.”
0:H median ≥ $65,000; :
a
H median < $65,000 (claim)
(b) The critical value is 01.96.z=−
(c) x = 24
456 CHAPTER 11 │ NONPARAMETRIC TESTS
13. (a) The claim is “the median age of Twitter® users is greater than 30 years old.”
0:H median ≤ 30; :
a
H median > 30 (claim)
14. (a) The claim is “the median age of Facebook® users is less than 32 years old.”
0:H median ≥ 32; :
a
H median < 32 (claim)
(b) The critical value is 5.
15. (a) The claim is “the median number of rooms in renter-occupied units is four.”
0:H median = 4; (claim) :
a
H median ≠ 4
(b) The critical value is 01.96.z=−
16. (a) The claim is “the median square footage of renter-occupied units is 1350 square feet.”
0:H median = 1350; (claim) :
a
H median ≠ 1350
(b) The critical value is 5.
17. (a) The claim is “the median hourly wage of computer systems analysts is $37.06.”
0:H median = $37.06; (claim) :
a
H median ≠ $37.06
(b) The critical value is 02.575.z=−
(c) x = 18
CHAPTER 11 │ NONPARAMETRIC TESTS 457
18. (a) The claim is “the median hourly wage of podiatrists is at least $55.89.”
0:H median ≥ $55.89; (claim) :
a
H median < $55.89
(b) The critical value is 6.
19. (a) The claim is “the lower back pain intensity scores decreased after acupuncture.”
0:H The lower back pain intensity scores have not decreased.
:
a
H The lower back pain intensity scores have decreased. (claim)
(b) The critical value is 1.
20. (a) The claim is “the lower back pain intensity scores decreased after taking the anti-
inflammatory drugs.”
0:H The lower back pain intensity scores have not decreased.
21. (a) The claim is “the student’s critical reading SAT scores improved.”
0:H The SAT scores have not improved.
:
a
HThe SAT scores have improved. (claim)
(b) The critical value is 2.
(c) x = 4
(d) Fail to reject 0.H
(e) There is not enough evidence at the 5% level of significance to conclude that the critical
reading SAT scores improved.
22. (a) The claim is “the student’s critical reading SAT scores improved the second time they took
the SAT.”
0:H The SAT scores have not improved.
458 CHAPTER 11 │ NONPARAMETRIC TESTS
23. (a) The claim is “the proportion of adults who feel older than their real age is equal to the
proportion of adults who feel younger than their real age.”
0:H The proportion of adults who feel older than their real age is equal to the proportion
24. (a) The claim is “the proportion of adults who contact their parents by phone weekly is equal to
the proportion of adults who contact their parents by phone daily.”
0:H The proportion of adults who contact their parents by phone weekly is equal to the
proportion of adults who contact their parents by phone daily. (claim)
25. Hypothesis test results:
Parameter: median of variable
0
H: Parameter = 22.55
A
H: Parameter ≠ 22.55
Variable a n for test
Hourly wages (in dollars) 14 13
CHAPTER 11 │ NONPARAMETRIC TESTS 459
26. Hypothesis test results:
Parameter: median of variable
0
H: Parameter ≤ 57
A
H: Parameter > 57
Variable n n for test Sample Median
Age 24 21 50
27. (a) The claim is “the median weekly earnings of female workers are less than or equal to $638.”
0:H median ≤ $638; (claim) :
a
H median > $638
(b) The critical value is 02.33.z=
(c) x = 29
28. (a) The claim is “the median weekly earnings of male workers are greater than $798.”
0:H median ≤ $798; :
a
H median > $798 (claim)
(b) The critical value is 02.33.z=
(c) x = 45
29. (a) The claim is “the median age of brides at the time of their first marriage is less than or equal
to 26 years.”
0:H median ≤ 26; :
a
H median > 26 (claim)
(b) The critical value is 01.645.z=
(c) x = 35
460 CHAPTER 11 │ NONPARAMETRIC TESTS
(d) Fail to reject 0.H
(e) There is not enough evidence at the 5% level of significance to reject the counselor’s claim
that the median age of brides at the time of their first marriage is less than or equal to 26
years.
11.2 THE WILCOXON TESTS
11.2 Try It Yourself Solutions
1a. The claim is “a spray-on water repellant is effective.”
0:H There is no difference in the amounts of water repelled.
:
a
H There is a difference in the amounts of water repelled.
b. 0.01
α
=
c. n = 11
d. The critical value is 5.
e.
No
repellent
Repellent
applied
Difference
Absolute
value
Rank
Signed
rank
8 15
−7 7 11
−11
7 12
−5 5 9
−9
7 11
−4 4 7.5
−7.5
CHAPTER 11 │ NONPARAMETRIC TESTS 461
f. Fail to reject 0.H
g. There is not enough evidence at the 1% level of significance for the quality control inspector to
conclude that the spray-on water repellent is effective.
e.
Ordered data Sample Rank Ordered data Sample Rank
1.7 B 1 5.3 B 13
1.8 B 2 5.6 B 14
2.2 B 3 5.8 A 15
2.5 A 4 6.0 A 16
3.0 A 5.5 6.2 A 17
3.0 B 5.5 6.3 A 18
f. 11 2
(1)
12(12121)150
22
R
nn n
µ
++ ++
===
12 1 2
(1)
(12)(12)(12 12 1) 17.321
12 12
R
nn n n
σ
++ ++
== ≈
120.5 150 1.703
17.321
R
R
R
z
µ
σ
−−
=≈ ≈−
11.2 EXERCISE SOLUTIONS
1. If the samples are dependent, use the Wilcoxon signed-rank test. If the samples are independent,
use the Wilcoxon rank sum test.
3. (a) The claim is “there was no reduction in diastolic blood pressure.”
0:H There is no reduction in diastolic blood pressure. (claim)
:
a
H There is a reduction in diastolic blood pressure.
(b) Wilcoxon signed-rank test
(c) The critical value is 10.
(d)
Before treatment
After treatment
Difference Absolute
difference
Rank
Signed rank
108 99 9 9 8 8
109 115
−6 6 4.5
−4.5
120 105 15 15 12 12
129 116 13 13 10.5 10.5
Sum of negative ranks = −17
Sum of positive ranks = 61
17
s
w=
(e) Fail to reject 0.H
(f) There is not enough evidence at the 1% level of significance to reject the claim that there was
no reduction in diastolic blood pressure.
CHAPTER 11 │ NONPARAMETRIC TESTS 463
(d)
Ordered data Sample Rank
43 M 1
45 M 2
47 M 3
49 M 4
53 WT 5
55 WT 6.5
55 M 6.5
70 WT 20
R = sum ranks of Manufacturing = 83
11 2
(1)
10(10101)105
22
R
nn n
µ
++ ++
===
12 1 2
(1)
(10)(10)(10 10 1) 13.229
12 12
R
nn n n
σ
++ ++
== ≈
5. (a) The claim is “the cost of prescription drugs is lower in Canada than in the United States.”
0:H The cost of prescription drugs is not lower in Canada than in the United States.
:
a
H The cost of prescription drugs is lower in Canada than in the United States. (claim)
(b) Wilcoxon signed-rank test
(c) The critical value is 4.
464 CHAPTER 11 │ NONPARAMETRIC TESTS
(d)
Cost in US. Cost in Canada Difference Absolute
difference
Rank Signed rank
1.26 1.04 0.22 0.22 1 1
1.76 0.82 0.94 0.94 3 3
4.19 2.22 1.97 1.97 7 7
3.36 2.22 1.14 1.14 4 4
that the cost of prescription drugs is lower in Canada than in the United States.
6. (a) The claim is “there is a difference in the earnings of people with bachelor’s degrees and those
with associate’s degrees.”
0:H There is no difference in the earnings.
:
a
H There is a difference in the earnings. (claim)
(b) Wilcoxon rank sum test
(c) The critical values are 01.96.z=±
45 A 9.5
47 A 11
50 B 12.5
50 B 12.5
CHAPTER 11 │ NONPARAMETRIC TESTS 465
11 2
(1)
10(10 11 1) 110
22
R
nn n
µ
++ ++
===
7. (a) The claim is “there is a difference in the salaries earned by teachers in Wisconsin and
Michigan.”
0:H There is not a difference in the salaries.
:
a
H There is a difference in the salaries. (claim)
(d)
Ordered data Sample Rank
46 WI 1
49 MI 2
50 WI 3.5
50 WI 3.5
51 WI 6.5
51 WI 6.5
51 WI 6.5
51 MI 6.5
72 MI 23
R = sum ranks of Wisconsin = 100.5
11 2
(1)
11(11 12 1) 132
22
R
nn n
µ
++ ++
===
466 CHAPTER 11 │ NONPARAMETRIC TESTS
100.5 132 1.94
16.248
R
R
R
z
µ
σ
−−
=≈ ≈−
8. (a) The claim is “the experimental medication affects an individual’s heart rate.”
0:H The experimental medication does not affect an individual’s heart rate.
:
a
H The experimental medication does affects an individual’s heart rate. (claim)
(b) Wilcoxon signed-rank test
(c) The critical value is 25.
(d)
Before After Difference Absolute value Rank Signed rank
72 73 −1 1 1.5
−1.5
81 80 1 1 1.5 1.5
75 75 0 0 — —
Sum of negative ranks = −46.5
Sum of positive ranks = 31.5
31.5
s
w=
(e) Fail to reject 0.H
(f) There is not enough evidence at the 5% level of significance for the physician to conclude
that the experimental medication affects an individual’s heart rate.
CHAPTER 11 │ NONPARAMETRIC TESTS 467
Before After Difference Absolute value Rank Signed rank
36.4 36.7 −0.3 0.3 4.5
−4.5
36.4 36.9 −0.5 0.5 11
−11
36.6 37.0 −0.4 0.4 7
−7
36.6 37.5 −0.9 0.9 17
−17
36.7 36.3 0.4 0.4 7 7
37.5 38.9 −1.4 1.4 25
−25
37.6 39.0 −1.4 1.4 25
−25
37.8 39.1 −1.3 1.3 21.5
−21.5
37.9 39.4 −1.5 1.5 28.5
−28.5
37.9 39.4 −1.5 1.5 28.5
−28.5
38.1 39.5 −1.4 1.4 25
−25
33.6 34.2 −0.6 0.6 14.5
−14.5
34.2 34.7 −0.5 0.5 11
−11
35.1 34.9 0.2 0.2 2.5 2.5
35.2 34.9 0.3 0.3 4.5 4.5
35.3 35.3 0 0 — —
35.5 35.9 −0.4 0.4 7
−7
35.9 36.4 −0.5 0.5 11
−11
[]
( 1) 32(32 1)
43.5 220.5
44
4.123
( 1)(2 1) 2860
32(32 1) (2)32 1
24 24
s
nn
w
znn n
++
−−
−
== ==−
++ ++
468 CHAPTER 11 │ NONPARAMETRIC TESTS
10. The claim is “a fuel additive improves gas mileage.”
0:H The fuel additive does not improve gas mileage.
:
a
H The fuel additive does improve gas mileage. (claim)
The critical value is 01.645.z=−
Before After Difference Absolute value Rank Signed rank
34.0 36.6 −2.6 2.6 24.5
−24.5
34.2 36.7 −2.5 2.5 21.5
−21.5
34.4 37.2 −2.8 2.8 29.5
−29.5
34.4 37.2 −2.8 2.8 29.5
−29.5
32.3 34.9 −2.6 2.6 24.5
−24.5
33.0 34.9 −1.9 1.9 11.5
−11/5
33.1 35.7 −2.6 2.6 24.5
−24.5
33.7 36.0 −2.3 2.3 20
−20
33.7 36.2 −2.5 2.5 21.5
−21.5
33.8 36.5 −2.7 2.7 27.5
37.1 38.8 −1.7 1.7 6
−6
37.1 38.9 −1.8 1.8 8.5
−8.5
37.2 39.1 −1.9 1.9 11.5
−11.5
37.9 39.1 −1.2 1.2 2
−2
37.9 39.2 −1.3 1.3 3
−3
Sum of negative ranks = −528
Sum of positive ranks = 0
0
s
w=
CHAPTER 11 │ NONPARAMETRIC TESTS 469
Reject
0.H There is enough evidence at the 5% level of significance to support the engineer’s
claim that the fuel additive improves gas mileage.
11.3 THE KRUSKAL-WALLIS TEST
11.3 Try It Yourself Solutions
1a. The claim is “the distribution of the veterinarians’ salaries in these three states are different.”
e.
Ordered data State Rank Ordered data State Rank
88.28 CA 1 99.70 CA 15
88.80 PA 2 99.75 NY 16
92.50 NY 3 99.95 CA 17
93.10 NY 4 99.99 PA 18
94.40 PA 5 100.55 PA 19
95.15 CA 6 100.75 CA 20
99.20 PA 14 113.90 CA 28
1
R
= 157.5
2
R
= 119
3
R
= 129
f.
222
3
12
123
12 3( 1)
(1)
RRR
HN
NN n n n
⎛⎞
=++−+
⎜⎟
+⎝⎠
470 CHAPTER 11 │ NONPARAMETRIC TESTS
11.3 EXERCISE SOLUTIONS
1. The conditions for using a Kruskal-Wallis test are that each sample must be randomly selected
and the size of each sample must be at least 5.
CHAPTER 11 │ NONPARAMETRIC TESTS 471
(c)
Ordered data Sample Rank
535 VA 1
546 VA 2
618 VA 3
947 CT 14
980 MA 15
1025 MA 16
1040 CT 17
1105 MA 18
1110 MA 19
1165 CT 20
1295 MA 21
4. The claim is “the distributions of the registered nurses’ hourly pay rates in these three states are
different.”
0:H There is no difference in the hourly pay rates.
:
a
H There is a difference in the hourly pay rates. (claim)
(b) The critical value is 5.991.
472 CHAPTER 11 │ NONPARAMETRIC TESTS
(c)
Ordered data Sample Rank
24.6 KT 1
25.1 IN 2
25.25 OH 3
25.58 KT 4
25.75 OH 5
28.25 IN 12
28.55 KT 13
29.44 IN 14
30.15 OH 15
30.2 KT 16
30.24 IN 17
31.55 OH 18
0.395
≈
(d) Fail to reject 0.H
(e) There is not enough evidence at the 5% level of significance to conclude that the distributions
of the registered nurses’ hourly pay rates in the three states are different.
5. The claim is “the distributions of the annual salaries in these four states are different.”